An Upper Bound of the Bezout Number for Piecewise Algebraic Curves over a Rectangular Partition
A piecewise algebraic curve is a curve defined by the zero set of a bivariate spline function. Given two bivariate spline spaces (Δ) and (Δ) over a domain D with a partition Δ, the Bezout number BN(m,r;n,t;Δ) is defined as the maximum finite number of the common intersection points of two arbitrary piecewise algebraic curves and , where ∈ (Δ) and ∈ (Δ). In this paper, an upper bound of the Bezout number for piecewise algebraic curves over a rectangular partition is obtained.
Let be a bounded domain, and let be the collection of real bivariate polynomials with total degree not greater than . Divide by using finite number of irreducible algebraic curves we get a partition denoted by . The subdomains are called the cells. The line segments that form the boundary of each cell are called the edges. Intersection points of the edges are called the vertices. The vertices in the inner of the domain are called interior vertices, otherwise are called boundary vertices. For a vertex , its so-called star means the union of all cells in sharing as a common vertex, and its degree is defined as the number of the edges sharing as a common endpoint. If is odd, we call an odd vertex. For integers and with , the bivariate spline space with degree and smoothness over with respect to is defined as follows [1, 2]: For a spline , the zero set is called a piecewise algebraic curve [1, 2]. Obviously, the piecewise algebraic curve is a generalization of the usual algebraic curve [3, 4].
In fact, the definition of piecewise algebraic curve is originally introduced by Wang in the study of bivariate spline interpolation. He pointed out that the given interpolation knots are properly posed if and only if they are not lying on a same nonzero piecewise algebraic curve [1, 2]. Hence, to solve a bivariate spline interpolation problem, it is necessary to deal with the properties of piecewise algebraic curve. Moreover, piecewise algebraic curve is also helpful for us to study the usual algebraic curve. Besides, piecewise algebraic curve also relates to the remarkable Four-Color conjecture [5–7]. In fact, the Four-Color conjecture holds if and only if, for any triangulation, there exist three linear piecewise algebraic curves such that the union of them equals the union of all central lines of all triangles in the triangulation. We know that any triangulation is 2-vertex signed [6, 7], which means the vertices of the triangulation can be marked by or such that the vertices of every triangle in the triangulation are marked by different numbers. So, we remark that the Four-Color conjecture holds if and only if there exist three nonequivalent mark methods. Based on these observations, in a word, piecewise algebraic curve is a new and important topic of computational geometry and algebraic geometry. It is of important theoretical and practical significance in many fields such as bivariate spline interpolation and computer-aided geometric design (CAGD). Hence, it is necessary to continue to study it.
In this paper, we mainly focus on the Bezout number for piecewise algebraic curves. It is well known that the Bezout theorem for usual algebraic curves is very important in algebraic geometry [3, 4]. Its weak form says that two algebraic curves with degree and , respectively, will have infinite number of common intersection points if they have more intersection points (including multiple points) than the product of their degrees, that is, the so-called Bezout number. Similarly, for two given bivariate spline spaces and , the following number  is called the Bezout number, where denotes the number of the common intersection points (including multiple points) of and . It also implies that two piecewise algebraic curves determined by two splines in and respectively will have infinite number of common intersection points if they have more intersection points than . Needless to say, it is crucial for us to obtain . Obviously, if , then we have However, we remark that it is very hard to obtain exact . On the one hand, piecewise algebraic curve itself is difficult; on the other hand, the Bezout number is also complicated; we know it not only relies on the degrees , and the smoothness orders , , but also the dimensions of and and the geometric characteristics of as well [6, 7]. Hence, we can only get an upper bound for sometimes.
Throughout the relative literatures, we find that there have been some results on the Bezout number for piecewise algebraic curves over triangulations [6–9], while this paper, as a continue paper to fill a gap, is mainly devoted to an upper bound of the Bezout number for two piecewise algebraic curves over a rectangular partition. Our method is different from the methods in [6–9] and is new and effective. The remainder is organized as follows. In Section 2, for the sake of integrity, some prevenient results on the Bezout number for piecewise algebraic curves over parallel lines partition and triangulation are introduced; Section 3 is the main section, in this section; assume to be a rectangular partition, an upper bound of is well estimated; finally, this paper is concluded in Section 4 with a conjecture.
Denoted by the partition containing only parallel lines (Figure 1), we have the followeing theorem.
Theorem 2.1 (see ). One has
For a triangulation , the Bezout number for two continuous piecewise algebraic curves satisfying the following theorem.
Theorem 2.2 (see ). Consider where and is the number of the triangles and the odd interior vertices in , respectively, and means the maximum integer not greater than .
Theorem 2.3 (see ). If is an interior vertex, then and if is a boundary vertex, then
Based on Theorem 2.3, by a combinatorial optimization method, a result for a general nonobtuse triangulation is obtained.
Theorem 2.4 (see ). For any given nonobtuse triangulation , where , , , and is the number of the triangles, the interior edges, the interior vertices, and the boundary vertices in .
We know that the Bezout number for piecewise algebraic curves over stars is the key issue for the Bezout number for piecewise algebraic curves over a general partition. In order to get , Wang and Xu generalized the smoothness order of Theorem 2.3 from to .
Theorem 2.5 (see ). For any nonobtuse star , if is an interior vertex, then and if is a boundary vertex, then
Here, we note that if is a boundary vertex, then , so holds unconditionally; hence we have
Recently, by using the theory of resultants, polar coordinates and periodic trigonometric spline, Theorem 2.5 is improved greatly when is an interior vertex.
Here, we also remark that if is a boundary vertex, then the second formula is likewise equivalent to By Theorem 2.6, we get a better upper bound of as follows.
Theorem 2.7. For any nonobtuse triangulation , one has where stands for the number of the odd interior vertices in .
Proof. Suppose that the number of the common intersection points of two piecewise algebraic curves and is finite and equals , where and . For a vertex in , let be the number of the common intersection points of and in . Summing for each vertex, so every common intersection point is counted triply. Hence, we get By Theorem 2.6, we have where By the Euler formulae and the equations , if is even, we get Similarly, if is odd, we have
In the next section, we will apply Theorems 2.1 and 2.6 to a rectangular partition. A good upper bound of the Bezout number for piecewise algebraic curves over a rectangular partition is derived, which fills a gap in the study of the Bezout number for piecewise algebraic curves over any partition.
Without loss of generality, let integers . For a rectangular domain , subdivide it into subrectangular cells by vertical lines and horizontal lines, and denote by the partition; see Figure 4 for an example. Let , , and be the collection of the interior vertices (), the boundary vertices that lying in the interior of the boundary edges (), and the else four corner vertices (), respectively. The cardinality of , , and is , , and , respectively. If a cell lies in the th column (from left to right) and in the th row (from bottom to top), we denote it by , ; . For an interior vertex , if it is the intersection point of the th vertical interior line and the th horizontal interior line, then we denote it by ; . The else boundary vertices can also be denoted similarly.
Assume that the number of the common intersection points of two piecewise algebraic curves and is finite and equals , where and . If , then is a partition containing only parallel lines essentially (Figure 1); by Theorem 2.1, we have In the rest of this section, assume .
3.1. The First Method
For a vertex in , let be the number of the common intersection points of and in . Summing for each vertex, so that every common intersection point is counted fourfold. Hence, we get By Theorem 2.6, we have where We have Let .
3.2. The Second Method
Let be the number of the common intersection points of and in the cell summing for all cells, we get
For the cells in the th row, by Theorem 2.1, we have Hence, Similarly, Let , . Considering , we have . This tells us a better summation method results in a better upper bound.
3.3. The Third Method
In this subsection, we will derive a rather better upper bound of than , , and by using a mixed method based on Theorems 2.1 and 2.6. We prefer this summation method.(1)If and are both even, then ; so (2)If is odd and is even, then ; so we have (3)If is even and is odd, then ; so (4)If and are both odd, then , hence
Theorem 3.1. Let be a rectangular partition, then
Let denotes the upper bound given in Theorem 3.1. We remark that is better than , , and . Since and , we only give the comparisons between and . For fixed , , and , we have the following results; see Table 1. In a word, we have .
In this paper, we mainly derive an upper bound for the Bezout number . The results of Theorem 3.1 are excellent than , , and . It is very useful in the fields of CAGD. For example, we are frequently needed to get the common intersection points of two piecewise algebraic curves [11, 12]. By Theorem 3.1, a prior estimation of the number of the common intersection points of two piecewise algebraic curves over a rectangular partition will be obtained. In future, we will try best to improve Theorem 3.1 to get the exact number or a lower upper bound for . In order to attract readers’ interest, here, we also give a conjecture on the Bezout number .
Conjecture 4.1. Consider If this conjecture holds, it can be applied into the study of the Nöther-type theorem [13, 14] and the Cayley-Bacharach theorem  for piecewise algebraic curves over rectangular partition.
This work was supported by the Fundamental Research Funds for the Central Universities (no. 201113037). The author appreciate the reviewers and editors for their careful reading, valuable suggestions, timely review, and reply.
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